As discussed in the introduction, [1]} develop an exact Bernoulli likelihood for the population augmented by the convenience sample. This approach specifies indicator \(R_{i} = 1\)  if unit \(i\)  is in the convenience sample or 0 if it is the finite population and develops an associated propensity score, \(\pi _{Ri} = \pi _{ci}/(\pi _{ci} + 1)\) . This expression is a special case of our main method in Section  where one-arm is the convenience sample and the other arm is the entire population. We derive this expression that we label our one-arm method in Section . So the exact likelihood specified in [1]} is a special case of our main method under a one-arm sample set-up. As with [3]}, they approximate their log-likelihood on the observed sample with,
\(\ell (\mathbf {\theta }_{c}) = \mathop {\sum }_{i \in S_{c}}\log \left(\pi _{Ri}(\mathbf {x}_{i},\mathbf {\theta }_{c})\right) + \mathop {\sum }_{i \in S_{r}}d_{ri}\log \left(1-\pi _{Ri}(\mathbf {x}_{i},\mathbf {\theta }_{c})\right).\) 
